Properties

Label 16-2736e8-1.1-c1e8-0-4
Degree $16$
Conductor $3.140\times 10^{27}$
Sign $1$
Analytic cond. $5.18973\times 10^{10}$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·13-s + 48·31-s − 24·43-s + 52·49-s + 12·61-s + 24·67-s − 20·73-s − 24·79-s − 48·103-s − 48·109-s + 48·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3.32·13-s + 8.62·31-s − 3.65·43-s + 52/7·49-s + 1.53·61-s + 2.93·67-s − 2.34·73-s − 2.70·79-s − 4.72·103-s − 4.59·109-s + 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(5.18973\times 10^{10}\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 3^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.917021798\)
\(L(\frac12)\) \(\approx\) \(4.917021798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
good5 \( 1 - 2 T^{4} - 621 T^{8} - 2 p^{4} T^{12} + p^{8} T^{16} \)
7 \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 24 T^{2} + 338 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 2 T + p T^{2} )^{4}( 1 + 5 T + p T^{2} )^{4} \)
17 \( 1 - 36 T^{2} + 26 p T^{4} - 9936 T^{6} + 271251 T^{8} - 9936 p^{2} T^{10} + 26 p^{5} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \)
23 \( 1 + 48 T^{2} + 1102 T^{4} + 6912 T^{6} - 53853 T^{8} + 6912 p^{2} T^{10} + 1102 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} \)
29 \( 1 + 20 T^{2} - 950 T^{4} - 6640 T^{6} + 778819 T^{8} - 6640 p^{2} T^{10} - 950 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \)
31 \( ( 1 - 12 T + 95 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
41 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + 12 T + 137 T^{2} + 1068 T^{3} + 8136 T^{4} + 1068 p T^{5} + 137 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 60 T^{2} - 950 T^{4} + 7920 T^{6} + 7870419 T^{8} + 7920 p^{2} T^{10} - 950 p^{4} T^{12} + 60 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 + 80 T^{2} + 3070 T^{4} - 183040 T^{6} - 14779181 T^{8} - 183040 p^{2} T^{10} + 3070 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \)
59 \( 1 - 104 T^{2} + 5038 T^{4} + 123136 T^{6} - 16382573 T^{8} + 123136 p^{2} T^{10} + 5038 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 - 6 T - 47 T^{2} + 234 T^{3} + 972 T^{4} + 234 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 12 T + T^{2} - 108 T^{3} + 6312 T^{4} - 108 p T^{5} + p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( 1 + 4 T^{2} - 6182 T^{4} - 15536 T^{6} + 12922867 T^{8} - 15536 p^{2} T^{10} - 6182 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \)
73 \( ( 1 + 10 T + 37 T^{2} - 830 T^{3} - 8660 T^{4} - 830 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 12 T + 25 T^{2} - 468 T^{3} - 3456 T^{4} - 468 p T^{5} + 25 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 204 T^{2} + 20294 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 + 176 T^{2} + 11278 T^{4} + 678656 T^{6} + 79908067 T^{8} + 678656 p^{2} T^{10} + 11278 p^{4} T^{12} + 176 p^{6} T^{14} + p^{8} T^{16} \)
97 \( ( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.82123654692791221233618695667, −3.60351297713783029901035087925, −3.18260780344351973077102111092, −3.13066525209632112361396126562, −3.08842471403192934616034397748, −3.03405548518104319931332782983, −2.99541298316773467735873320397, −2.76665942231057952209092302400, −2.62408345306410583412939530850, −2.56860293071202828999597214576, −2.55116620087242608614297425554, −2.54039909421324893081337640876, −2.17498707836844012099369225019, −2.10974245170766529226545453993, −1.88894658256382105155266900655, −1.85632210600389058134530562787, −1.73010470610953023238724493205, −1.48266034629287468277542400335, −1.12533354874520391151623710739, −1.07118789338785900002740522456, −0.911026782067512745171915955972, −0.69742773693096301001568402172, −0.64481624553379962603223412094, −0.57640784156064528213070809207, −0.15614822070705414483605247351, 0.15614822070705414483605247351, 0.57640784156064528213070809207, 0.64481624553379962603223412094, 0.69742773693096301001568402172, 0.911026782067512745171915955972, 1.07118789338785900002740522456, 1.12533354874520391151623710739, 1.48266034629287468277542400335, 1.73010470610953023238724493205, 1.85632210600389058134530562787, 1.88894658256382105155266900655, 2.10974245170766529226545453993, 2.17498707836844012099369225019, 2.54039909421324893081337640876, 2.55116620087242608614297425554, 2.56860293071202828999597214576, 2.62408345306410583412939530850, 2.76665942231057952209092302400, 2.99541298316773467735873320397, 3.03405548518104319931332782983, 3.08842471403192934616034397748, 3.13066525209632112361396126562, 3.18260780344351973077102111092, 3.60351297713783029901035087925, 3.82123654692791221233618695667

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.